Answer
$8\log_2{a}+2\log_2{b}$
Work Step by Step
The given expression is equivalent to:
\begin{align*}
&=\log_2{\left((a^2)^4(\sqrt{b})^4\right)}\\
&=\log_2{(a^8b^2)}
\end{align*}
Recall:
(1) $\log_a{(mn)}=\log_a{m} + \log_a{n}$
(2) $\log_a{\left(\frac{m}{n}\right)}=\log_a{m} - \log_a{n}$
(3) $\log_a{a^m}=m\log_a{m}$
Use rule (1) above to obtain:
\begin{align*}
\log_2{\left(a^8b^2\right)}&=\log_2{\left(a^8\right)}+\log_2{(b^2)}
\end{align*}
Use rule (3) above to obtain:
\begin{align*}
\log_2{(a^8)}+\log_2{(b^2)}&=8\log_2{a}+2\log_2{b}
\end{align*}
